The general goal of the research plan described in this application is to develop methods for studying a class of statistical problems arising from biomedical and psychiatric clinical trials in which the response is multivariate and the parameters obey an order relationship. The covariance structures in such situations and the ordering relation can often be used to substantially improve the statistical inference. The basic approach is (i) to formulate a single multivariate hypothesis that captures the most relevant question, (ii) to develop testing procedures, (iii) to invert testing procedures to provide confidence regions, (iv) to develop procedures that allow univariate testing to be done following the multivariate testing, and (v) to provide algorithms for the implementation of (i) to (iv). Illustrations of the kinds of experiments we will consider are two-arm clinical trials with multiple endpoints, new drug screening trials in which several dose levels are compared to the zero dose control, and combination-drug trials for testing whether each component contributes to the effect. For each statistical problem we will develop normal theory applicable to large samples. Specifically, we will derive the LR test and invert it to obtain a confidence region. For small samples, non-parametric statistics will be developed using multivariate U-statistics. However, to use such statistics may require asymptotic distribution theory, which is usually in a normality framework. Thus, the normal theory that is developed for large samples may also be useful for small samples. The robustness of the normality assumption will be examined through simulations based on other possible distributions. For each test obtained the power will be studied and critical values tabulated. Practical ways for displaying or describing a high-dimensional confidence region will be sought.